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# Illustration 16.6: The Fourier Series, Quantitative Features

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Fourier's theorem states that any periodic function (whether periodic in position or time) can be represented as a sum of sine waves. We find that sometimes we may need an infinite number of sines, but nonetheless we may describe any periodic phenomena this way. In this animation we investigate odd periodic functions in position with Fourier's theorem. Restart.

ANY odd periodic function of x (a period of L between 0 and L as opposed to between -L/2 to L/2) can be described in terms of a Fourier series as:

f(x) = Σ A_{n} sin (n*2*π*x/L),

where in this animation L = 1. A_{n} is the result of an integral that represents the overlap between the original function and a particular Fourier component (one term in the Fourier series represented by the integer n). In order to get this to exactly work out, there must be a 2/L (in our case just a factor of 2 since L = 1 here) included in the integral. Verify that this is necessary by predicting A_{3} for the function sin(3*2*pi*x) and then use the animation as a check.

*Remember to use the proper syntax, such as -10+0.5*t, -10+0.5*t*t, and -10+0.5*t^2. Revisit Exploration 1.3 to refresh your memory.*

Try various odd functions to see the result of the integral, A_{n}. Consider the following functions (you may copy and paste them in directly):

the Sawtooth Wave in Illustration 16.5(1-2*x)*step(1-x)+(1-2*(x-1))*step(x-1) |
the Square Wave in Illustration 16.5step(0.5-x)-step(x-0.5)*step(1-x)+step(x-1) |

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